The notion of the density of states originated in solid state physics [1]} where it is, roughly speaking, the number of states (eigenfunctions) per unit volume whose energy (eigenvalue) lies in a given range.
This notion is also of interest in mathematics due to its relation to \(l^2\) -invariants in K-theory and geometry and the von Neumann dimension [2]} [3]} as well as the study of spectral measures of groups.
The study of spectra, spectral measures and density of states on infinite graphs is a large area of mathematics (see the survey [4]}).
